Is there a quick way to exit or restart a mission that you are already replaying?

When replaying a mission for gold medal, you sometimes know early that you won't get it, yet you have to play it to the end, find some way to fail (and then you only restart from the latest checkpoint) or reload your story game.

Is there some shortcut to just restart the mission from the beginning?

I'm not sure, since not on the Xbox at the moment. But did you try to go to the system settings when you press start and load the game?
If that is no help, maybe you should just quit to the dashboard and restart the game. It's tedious, I know. But when the autosave feature is turned on, everything before the mission will be 'back' again (stats-wise), but your current character might be doing something else than you did last time.

For example: In your previous (selfmade with the phone or home) save, you were in a black car. Now you do the method written above.
Suddenly, you are in a rollercoaster, wearing different things.

Everything else (money, time, progress, stats) stays the same as just before you started the mission, or: when it autosaved the last time.

i have been doing a quick save before the missions all tho sometimes just getting close to the waypoint sets it off so save ahead of the waypoint. If saved before tho just reload game save and you'll be exactly where you saved.